Kirkman covering designs with even-sized holes

A Kirkman holey covering design, denoted by KHCD(gu)(gu), is a resolvable group-divisible covering design of type gugu. Each of its parallel class contains one block of size δδ, while other blocks have size 3. Here δδ is equal to 2, 3 and 4 when gu≡2gu≡2, 3 and 4 (mod 3) in turn. In this paper, we study the existence problem of a KHCD(gu)(gu) which has minimum possible number of parallel classes, and give a solution for most values of even gg and uu.